zbMATH — the first resource for mathematics

A generic time hierarchy with one bit of advice. (English) Zbl 1173.68023
Summary: We show that for any reasonable semantic model of computation and for any positive integer \(a\) and rationals \(1 \leq c < d\), there exists a language computable in time \(n^d\) with \(a\) bits of advice but not in time \(n^c\) with \(a\) bits of advice. Our result implies the first such hierarchy theorem for randomized machines with zero-sided error, quantum machines with one- or zero-sided error, unambiguous machines, symmetric alternation, Arthur-Merlin games of any signature, etc. Our argument yields considerably simpler proofs of known hierarchy theorems with one bit of advice for randomized and quantum machines with two-sided error.
Our paradigm also allows us to derive stronger separation results in which the machine with the smaller running time can receive more advice than the one with the larger running time. We present a unified way to derive such results for randomized and quantum machines with two-sided error and for randomized machines with one-sided error.

68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
68Q05 Models of computation (Turing machines, etc.) (MSC2010)
68Q10 Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.)
81P68 Quantum computation
Full Text: DOI